From [1]} ([1]}) (Theorem 2.1) we know that \(\Phi ({\bf R}_i^{T+1}) \le \Phi ({\bf R}_i^{T}) + C\)  whenever external or internal regret matching is played on iteration \(T+1\) . For external regret matching, \(C = \Delta ^2 |A_I|\) . For internal regret matching, \(C = \Delta ^2 |A_I|^2\) . This result holds regardless of the sequence of actions that took place up to \(T\) .
\(\sum _{i \in P} \Phi ({\bf R}^T_i + r^{T+1}_i) & \le |P|CT + |P|C \le |P|C(T+1) \hspace{28.45274pt} \\\sum _{i \in P} \frac{\Phi ({\bf R}^T_i + r^{T+1}_i)}{(T+1)^2} & \le \frac{|P|C}{T+1} \\\min _{w^{T+1}} \sum _{i \in P} \frac{\Phi ({\bf R}^T_i + w^{T+1} r^{T+1}_i)}{(T + w^{T+1})^2} & \le \frac{|P|C}{T + 1} \hspace{28.45274pt} \textrm {(w^{T+1} selected by Algorithm~\ref {mainalgorithm})} \\\sum _{i \in P} \frac{w^{\prime 2}\Phi ({\bf R}^T_i + w^{T+1} r^{T+1}_i)}{w^{\prime 2}(T + w^{T+1})^2} & \le \frac{|P|C}{T + 1} \hspace{28.45274pt} \textrm {Rescaling }w^{\prime } = \frac{T+1}{T+w^{T+1}}  \\\sum _{i \in P} \frac{\Phi (w^{\prime } {\bf R}^T_i + w^{\prime } w^{T+1} r^{T+1}_i)}{(T+1)^2} & \le \frac{|P|C}{T + 1} \\\sum _{i \in P} \Phi (w^{\prime } {\bf R}^T_i + w^{\prime } w^{T+1} r^{T+1}_i) & \le |P|C (T + 1) \\\) 
